Triangle Calculator

Triangles are classified by their sides and angles. Understanding these classifications helps you identify properties and choose the right solving approach.

Classification by Sides

Equilateral Triangle

• All three sides are equal (a = b = c)
• All three angles are equal (60 degrees each)
• Has three lines of symmetry
• Most "balanced" triangle possible
• Area formula: A = (sqrt(3)/4) * s^2 where s is the side length

Isosceles Triangle

• Two sides are equal (e.g., a = b, but c is different)
• Two angles are equal (the angles opposite the equal sides)
• Has one line of symmetry
• Common in architecture and design
• The unequal side is called the "base"

Scalene Triangle

• All three sides are different (a != b != c)
• All three angles are different
• No lines of symmetry
• Most general type of triangle
• Requires Law of Cosines or Law of Sines to solve

Classification by Angles

Acute Triangle

• All angles are less than 90 degrees
• The square of the longest side is less than the sum of squares of the other two sides
• All altitudes fall inside the triangle

Right Triangle

• One angle is exactly 90 degrees
• Follows the Pythagorean theorem: a^2 + b^2 = c^2 (where c is the hypotenuse)
• The side opposite the right angle is the longest (hypotenuse)
• Special cases: 45-45-90 and 30-60-90 triangles have predictable ratios

Obtuse Triangle

• One angle is greater than 90 degrees
• The square of the longest side is greater than the sum of squares of the other two sides
• One altitude falls outside the triangle

There are several ways to calculate the area of a triangle, depending on what information you have.

Base and Height Method

The most intuitive formula:

Area = (1/2) * base * height

Where:

• base is any side of the triangle
• height (altitude) is the perpendicular distance from the base to the opposite vertex

Example: A triangle with base 10 cm and height 6 cm:

• Area = (1/2) * 10 * 6 = 30 square cm

Heron's Formula (Three Sides Known)

When you know all three sides (SSS), use Heron's formula:

Area = sqrt(s * (s-a) * (s-b) * (s-c))

Where:

• s = (a + b + c) / 2 (the semi-perimeter)
• a, b, c are the three side lengths

Example: A triangle with sides 5, 6, and 7:

1. s = (5 + 6 + 7) / 2 = 9
2. Area = sqrt(9 * (9-5) * (9-6) * (9-7))
3. Area = sqrt(9 * 4 * 3 * 2) = sqrt(216) = 14.7 square units

Two Sides and Included Angle (SAS)

Area = (1/2) * a * b * sin(C)

Where:

• a and b are two sides
• C is the angle between them

Example: Two sides of 8 and 10 with an included angle of 60 degrees:

• Area = (1/2) * 8 * 10 * sin(60)
• Area = 40 * 0.866 = 34.64 square units

Coordinate Method

If you have the coordinates of all three vertices (x1,y1), (x2,y2), (x3,y3):

Area = (1/2) * |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|

This is useful in computer graphics and surveying applications.

Different combinations of known sides and angles require different solving methods. Here's a complete guide to each case.

SSS (Side-Side-Side)

Given: All three sides (a, b, c)

Method: Use the Law of Cosines to find angles

1. Find angle A: cos(A) = (b^2 + c^2 - a^2) / (2bc)
2. Find angle B: cos(B) = (a^2 + c^2 - b^2) / (2ac)
3. Find angle C: C = 180 - A - B (or use Law of Cosines)

Validity check: Each side must be less than the sum of the other two (triangle inequality).

SAS (Side-Angle-Side)

Given: Two sides and the included angle (e.g., a, b, and angle C)

Method: Use the Law of Cosines, then Law of Sines

1. Find side c: c^2 = a^2 + b^2 - 2ab*cos(C)
2. Find angle A: sin(A) = a*sin(C)/c
3. Find angle B: B = 180 - A - C

ASA (Angle-Side-Angle)

Given: Two angles and the included side (e.g., angles A, B, and side c)

Method: Use angle sum property, then Law of Sines

1. Find angle C: C = 180 - A - B
2. Find side a: a = c*sin(A)/sin(C)
3. Find side b: b = c*sin(B)/sin(C)

AAS (Angle-Angle-Side)

Given: Two angles and a non-included side (e.g., angles A, B, and side a)

Method: Similar to ASA

1. Find angle C: C = 180 - A - B
2. Find side b: b = a*sin(B)/sin(A)
3. Find side c: c = a*sin(C)/sin(A)

SSA (Side-Side-Angle) - The Ambiguous Case

Given: Two sides and a non-included angle (e.g., a, b, and angle A)

Caution: This case can have 0, 1, or 2 solutions!

Analysis:

• If A >= 90: One solution if a > b; no solution if a <= b
• If A < 90:
  • No solution if a < b*sin(A)
  • One solution if a = b*sin(A) (right triangle)
  • Two solutions if b*sin(A) < a < b
  • One solution if a >= b

The Law of Sines relates the sides of a triangle to the sines of their opposite angles. It's essential for solving triangles when you have angle-side combinations.

The Formula

a/sin(A) = b/sin(B) = c/sin(C) = 2R

Where:

• a, b, c are the side lengths
• A, B, C are the opposite angles
• R is the circumradius (radius of the circumscribed circle)

When to Use Law of Sines

Use the Law of Sines when you know:

• ASA: Two angles and the included side
• AAS: Two angles and any side
• SSA: Two sides and a non-included angle (with caution - ambiguous case)

Example Calculation

Given: Angle A = 40 degrees, Angle B = 60 degrees, side a = 10

1. Find angle C: C = 180 - 40 - 60 = 80 degrees

2. Find side b using Law of Sines:

   • b/sin(60) = 10/sin(40)
   • b = 10 * sin(60) / sin(40)
   • b = 10 * 0.866 / 0.643
   • b = 13.47

3. Find side c:

   • c = 10 * sin(80) / sin(40)
   • c = 10 * 0.985 / 0.643
   • c = 15.32

The Circumradius Connection

The Law of Sines also gives us the circumradius:

R = a / (2*sin(A))

This is the radius of the circle that passes through all three vertices of the triangle.

The Law of Cosines is a generalization of the Pythagorean theorem that works for all triangles. It's essential when you have SSS or SAS combinations.

The Formulas

c^2 = a^2 + b^2 - 2ab*cos(C) b^2 = a^2 + c^2 - 2ac*cos(B) a^2 = b^2 + c^2 - 2bc*cos(A)

Connection to Pythagorean Theorem

When angle C = 90 degrees:

• cos(90) = 0
• c^2 = a^2 + b^2 - 2ab*0 = a^2 + b^2

This is exactly the Pythagorean theorem! The Law of Cosines extends this to non-right triangles.

When to Use Law of Cosines

Use the Law of Cosines when you know:

• SSS: All three sides (to find angles)
• SAS: Two sides and the included angle (to find the third side)

Example 1: Finding a Side (SAS)

Given: a = 8, b = 6, angle C = 60 degrees

c^2 = 8^2 + 6^2 - 286cos(60) c^2 = 64 + 36 - 960.5 c^2 = 100 - 48 = 52 c = 7.21

Example 2: Finding an Angle (SSS)

Given: a = 5, b = 7, c = 8

To find angle A: cos(A) = (b^2 + c^2 - a^2) / (2bc) cos(A) = (49 + 64 - 25) / (278) cos(A) = 88 / 112 = 0.786 A = arccos(0.786) = 38.2 degrees

Determining Triangle Type

The Law of Cosines helps classify triangles:

• If c^2 = a^2 + b^2: Right triangle (cos(C) = 0, so C = 90 degrees)
• If c^2 < a^2 + b^2: Acute triangle (cos(C) > 0, so C < 90 degrees)
• If c^2 > a^2 + b^2: Obtuse triangle (cos(C) < 0, so C > 90 degrees)

One of the most fundamental properties of triangles is that the interior angles always sum to 180 degrees. This property is the foundation for solving many triangle problems.

The Fundamental Rule

Angle A + Angle B + Angle C = 180 degrees

This holds true for ALL triangles - right, acute, obtuse, equilateral, isosceles, and scalene.

Why Is This True?

Proof using parallel lines:

1. Draw a line through vertex A parallel to side BC
2. The alternate interior angles created equal angles B and C
3. These three angles at vertex A form a straight line (180 degrees)
4. Therefore: A + B + C = 180 degrees

Practical Applications

Finding the Third Angle

If you know two angles, you can always find the third:

• Example: If A = 45 degrees and B = 75 degrees, then C = 180 - 45 - 75 = 60 degrees

Checking Validity

If your calculated angles don't sum to 180 degrees, you have an error somewhere.

Understanding Constraints

• No angle can be >= 180 degrees
• No angle can be <= 0 degrees
• If one angle is 90 degrees (right triangle), the other two must sum to 90 degrees
• In an equilateral triangle, each angle is exactly 60 degrees

Exterior Angles

The exterior angle of a triangle (the angle formed by one side and the extension of an adjacent side) equals the sum of the two non-adjacent interior angles.

Exterior angle at C = A + B

This is another useful property for solving problems.

The height (or altitude) of a triangle is the perpendicular distance from a vertex to the opposite side (or its extension). Every triangle has three heights, one from each vertex.

Finding Height from Area

If you know the area and base:

height = (2 * Area) / base

This works because Area = (1/2) * base * height.

Finding Height Using Trigonometry

If you know a side and adjacent angle:

height_a = b * sin(C) = c * sin(B)

Where height_a is the altitude from vertex A to side a.

Height Formulas for Each Vertex

For a triangle with sides a, b, c and area A:

• h_a (height to side a) = 2A / a
• h_b (height to side b) = 2A / b
• h_c (height to side c) = 2A / c

Special Cases

Right Triangle

In a right triangle with the right angle at C:

• h_c (height to hypotenuse) = a*b/c
• The two legs themselves are altitudes to each other

Equilateral Triangle

All three heights are equal:

• h = (sqrt(3)/2) * s, where s is the side length
• h = s * 0.866 approximately

Isosceles Triangle

The height to the unequal side (base) bisects that side and the apex angle.

The Orthocenter

The three altitudes of a triangle always intersect at a single point called the orthocenter:

• In an acute triangle: inside the triangle
• In a right triangle: at the right angle vertex
• In an obtuse triangle: outside the triangle

Triangle calculations are essential in many practical fields. Here are some common applications where this calculator proves invaluable.

Construction and Architecture

Roof Design

• Calculate roof pitch and rafter lengths using right triangle math
• Determine the area of triangular gable ends
• Find the proper angle for drainage slope
• See our Roof Pitch Calculator for specialized roof calculations

Structural Support

• Triangles are the strongest geometric shape for structures
• Calculate forces in truss systems
• Determine brace lengths and angles
• Use our Rafter Calculator for construction projects

Land Surveying

• Triangulation is fundamental to surveying
• Calculate property areas with irregular boundaries
• Determine distances using angle measurements
• GPS systems use triangle calculations (trilateration)

Navigation

• Ships and aircraft use triangulation for position fixing
• Calculate course corrections and distances
• Determine bearing angles between waypoints

Engineering

• Force vector analysis often involves triangles
• Bridge design uses triangle principles extensively
• Mechanical linkages rely on triangle geometry

Art and Design

• The "rule of thirds" in photography uses implicit triangles
• Perspective drawing relies on triangular projections
• Logo design often incorporates triangle symbolism

Related Calculators

• Area Calculator - General area calculations for all shapes
• Pythagorean Theorem Calculator - Specialized for right triangles
• Roof Pitch Calculator - Calculate roof angles and slopes
• Rafter Calculator - Construction-specific calculations